Shapes of macromolecules in good solvents: field theoretical renormalization group approach
aa r X i v : . [ c ond - m a t . s o f t ] F e b Shapes of macromolecules in good solvents:field theoretical renormalization group approach ∗ V. Blavatska , C. von Ferber , Yu. Holovatch Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,1 Svientsitskii Str., 79011 Lviv, Ukraine Applied Mathematics Research Centre, Coventry University, CV1 5FB Coventry, UK Theoretische Polymerphysik, Universität Freiburg, 79104 Freiburg, Germany
Received May 11, 2011
In this paper, we show how the method of field theoretical renormalization group may be used to analyzeuniversal shape properties of long polymer chains in porous environment. So far such analytical calcula-tions were primarily focussed on the scaling exponents that govern conformational properties of polymermacromolecules. However, there are other observables that along with the scaling exponents are universal(i.e. independent of the chemical structure of macromolecules and of the solvent) and may be analyzed withinthe renormalization group approach. Here, we address the question of shape which is acquired by the longflexible polymer macromolecule when it is immersed in a solvent in the presence of a porous environment.This question is of relevance for understanding of the behavior of macromolecules in colloidal solutions, nearmicroporous membranes, and in cellular environment. To this end, we consider a previously suggested modelof polymers in d -dimensions [V. Blavats’ka, C. von Ferber, Yu. Holovatch, Phys. Rev. E, 2001, , 041102] inan environment with structural obstacles, characterized by a pair correlation function h ( r ) , that decays withdistance r according to a power law: h ( r ) ∼ r − a . We apply the field-theoretical renormalization group ap-proach and estimate the size ratio h R i / h R i and the asphericity ratio ˆ A d up to the first order of a double ε = 4 − d , δ = 4 − a expansion. Key words: polymer, quenched disorder, renormalization group
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1. Introduction
Polymer theory belongs to and uses methods of different fields of science: physics, physicalchemistry, chemistry, and material science being the principal ones. Historically, the structureof polymers remained under controversial discussion until, under the influence of the work byHermann Staudinger [1], the idea of long chain-like molecules became generally accepted. In thispaper we will concentrate on the universal properties of long polymer chains immersed in a goodsolvent, i.e. the properties that do not depend on the chemical structure of macromolecules and ofthe solvent. Usually a self-avoiding walk model is used to analyze such properties [2, 3]. At firstglance such a model is a rough caricature of a polymer macromolecule since out of its numerousinherent features it takes into account only its connectivity and the excluded volume modeled bya delta-like self-avoidance condition. However, it is widely recognized by now that the universalconformational properties of polymer macromolecules are perfectly described by the model of self-avoiding walks. It is instructive to note that the idea to describe polymers in terms of statisticalmechanics appeared already in early 30-ies due to Werner Kuhn [4] and already then enabledan understanding and a qualitative description of their properties. The present success in theiranalytic description which has lead to accurate quantitative results is to a large extent due to theapplication of field theoretic methods. In the pioneering papers by Pierre Gilles de Gennes and hisschool [2] an analogy was shown between the universal behaviour of spin systems near the critical ∗ A paper dedicated to Prof. Yurij Kalyuzhnyi on the occasion of his 60th birthday.c (cid:13)
V. Blavatska, C. von Ferber, Yu. Holovatch, 2011 . Blavatska, C. von Ferber, Yu. Holovatch point and the behaviour of long polymer macromolecules in a good solvent. In turn, this made itpossible to apply the methods of field theoretical renormalization group [5] to polymer theory.In spite of its success in explaining the universal properties of polymer macromolecules, theself-avoiding walk model does not encompass a variety of other polymer features. Different modelsand different methods are used for these purposes. In the context of this Festschrift it is appropriateto mention the approach based on the integral-equation techniques which is actively developed byYurij Kalyuzhnyi and his numerous colleagues [6]. In particular, this approach has enabled ananalytic description of chemically associating fluids and the representation of the most importantgeneric properties of certain classes of associating fluids [7]. It is our pleasure and honor to writea paper dedicated to Yurij Kalyuzhnyi on the occasion on his 60th birthday and doing so to wishhim many more years of fruitful scientific activity and to acknowledge our numerous commonexperiences in physics and not only therein.In this paper we will analyze the shapes of polymer macromolecules in a good solvent. Flexiblepolymer macromolecules in dilute solutions form crumpled coils with a global shape, which greatlydiffers from spherical symmetry and is surprisingly anisotropic, as it has been found experimentallyand confirmed in many analytical and numerical investigations [8–22]. Topological properties ofmacromolecules, such as the shape and size of a typical polymer chain configuration, are of interestin various respects. The shape of proteins affects their folding dynamics and motion in a celland is relevant in comprehending complex cellular phenomena, such as catalytic activity [23].The hydrodynamics of polymer fluids is essentially affected by the size and shape of individualmacromolecules [24]; the polymer shape plays an important role in determining its molecularweight in gel filtration chromatography [25]. Below we will show how the shape can be quantifiedwithin universal characteristics and how to calculate these characteristics analytically.The set up of this paper is as follows. In the next section we present some details of the first an-alytical attempt to study the shape of linear polymers in good solvent, performed by Kuhn in 1934.Since then, the study of topological properties of polymer macromolecules was developed, based ona mathematical description, which is presented in section 3 along with a short review of the knownresults for shape characteristics of flexible polymers. In section 4, we present details of the appli-cation of the field-theoretical renormalization group approach to the study of universal polymershape characteristics. Section 5 concerns the effect of structural disorder in the environment on theuniversal properties of polymer macromolecules. The model with long-range-correlated quencheddefects is exploited, and the shape characteristics are estimated in a field-theoretical approach. Weclose by giving conclusions and an outlook.
2. Shape of a flexible polymer: Kuhn’s intuitive approach
The subject of primary interest in this paper will be the shape of a long flexible polymermacromolecule. That is, assuming that a polymer coil constitutes of a large sequence of monomers,does this coil resemble a globe (which would be the naive expectation taken that each monomeris attached at random) or does its shape possess anisotropy. And, if it is anisotropic, what are theobservables to describe it quantitatively? In this section we start our analysis with a very simplemodel that allows us to make some quantitative conclusions. The aim of the calculations givenbelow is to explain how the anisotropy of the polymer shape arises already within a random walkmodel. That is to show that the anisotropy is essentially not an excluded volume effect (althoughits strength is effected by the excluded volume interaction as we will show in this paper) but ratherit is an intrinsic property arising from random walk statistics.The analysis given below is inspired by Kuhn’s seminal paper [4]. However, Kuhn’s explanationis based on combinatorial analysis and application of Stirling formula, while here we suggest aderivation based on the application of Bayesian probability [26]. Let us consider the so-calledGaussian freely jointed chain model consisting of N connected bonds capable of pointing in anydirection independently of each other. Any typical configuration of such a chain can be representedby the set of bond vectors { ~a n } , n = 1 , . . . , N , such that: h ~a i i = a = dℓ , (2.1) hapes of macromolecules in good solvents: field theoretical renormalization group approach (a) N/2N/4 0N r r r r(0) (b) r(T) Figure 1. a) Schematic presentation of a polymer chain with the end-to-end distance (positionvector of N th monomer) denoted as ~r . The component of the position vector of N/ th monomerin direction perpendicular to ~r is denoted by ~r . Vector ~r denotes the component of positionvector of N/ th monomer in direction perpendicular to both ~r and ~r . b) In the Edwardscontinuous chain model, the polymer is represented by a path ~r ( t ) parameterized by t T (see section 4). here d is the space dimension and the angular brackets stand for averaging with respect to differentpossible orientations of each bond. Fixing the starting point of the chain at the origin, one gets forthe end-to-end distance ~R e and its mean square: ~R e = N X i =1 ~a i , h ~R i = N a . (2.2)To get the second relation, one has to take into account that h ~a i ~a j i = 0 for i = j , since randomvariables ~a i are uncorrelated. Due to the central limit theorem, the distribution function of therandom variable R e = | ~R e | takes on a Gaussian form: P ( R e ) = (cid:16) d π h R i (cid:17) d/ e − dR h R i . (2.3)Numerical factors in (2.3) can be readily obtained from the normalization conditions for the distri-bution function and its second moment. Let us consider the three-dimensional ( d = 3 ) continuouschain, such that P ( R e )d R e = (cid:16) πN ℓ (cid:17) / e − R Nℓ πR d R e , (2.4)defines the probability that the end point of the chain is located in the interval between R e and R e + d R e . Following Kuhn, let us take the mean value of R e as one of the shape characteristics ofa chain and denote it by r ≡ h R e i . One gets for its value: r ≡ ∞ Z R e P ( R e )d R e = 2 ℓ r Nπ . (2.5)To introduce two more observables that will characterize the shape of the chain, let us proceedas follows. Having defined the end-to-end vector ~R e , let us point the z -axis along this vector (seefigure 1 (a)). Now, since both the starting and the end points of the chain belong to the z -axis, theprojection of the polymer coil on the xy -plane has a form of a loop. It is natural to expect thatthe largest deviation of a point on this loop from the origin corresponds to the N/ th step. Letus find the distance in the plane from the origin to this point (we will denote it by r hereafter)and take it as another shape characteristics of the chain. Note, that vector ~r lies in the xy -plainand therefore is two-dimensional. To do so, we have to find the distribution function of a position . Blavatska, C. von Ferber, Yu. Holovatch vector of a point on a loop. It is convenient to find such a distribution using the Bayes theoremfor conditional probability [26]. The theorem relates the conditional and marginal probabilities ofevents A and B , provided that the probability of B does not equal zero: P ( A | B ) = P ( B | A ) P ( A ) P ( B ) . (2.6)In our case, P ( A | B ) ≡ P ( r ) is the probability that the coordinate of the chain after N/ th step isgiven by a (now two-dimensional) vector ~r under the condition, that after N steps its coordinate is r = 0 . Then, P ( B | A ) is the probability for the chain that begins at the point with the coordinate ~r after N/ steps to return to the origin. Correspondingly, the prior probability P ( A ) is theprobability that the coordinate of the chain after N/ th step is given by a vector ~r (it is the so-called “unconditional” or “marginal” probability of A, in our case it is given by equation (2.3) for d = 2 ). P ( B ) is the prior or marginal probability of B, in our case this is the probability for thechain that starts at the origin to return back in N steps, i.e. a probability of a loop of N steps.Realizing that for our case P ( A ) = P ( B | A ) , that is probabilities to reach point ~r starting from theorigin is equal to the probability to reach the origin starting from the point ~r we get: P ( r ) = P ( A ) /P ( B ) , (2.7)where P ( A ) is given by equation (2.3) for d = 2 and P ( B ) may be found from the normalizationcondition. In the continuous chain representation we get for the probability that the N/ th bondof the chain is located in the interval between r and r + d r : P ( r )d r = 12 πN/ ℓ P ( B ) e − r N/ ℓ e − r N/ ℓ πr d r. (2.8)The mean value r follows: r ≡ ∞ Z rP ( r )d r = 4 N ℓ ∞ Z e − r Nℓ r d r = ℓ r πN . (2.9)Now, let us point an x -axis along ~r (again see figure 1 (a)) and repeat the above reasoningsconcerning the maximal distance in y -coordinate, i.e. concerning the y -coordinate of the N/ th stepof the chain. We will denote it by r . The result readily follows by the analogy with equation (2.9)taking into account that now the radius-vector is one-dimensional: r ≡ ∞ Z rP ( r )d r = (cid:16) πN ℓ (cid:17) / ∞ Z e − r Nℓ r d r = ℓ r Nπ . (2.10)Comparing (2.5), (2.9), and (2.10) one concludes that the shape of the chain is characterizedby three unequal sizes r , r , and r with the following relations: r r = 8 π ≃ . , r r = 4 √ ≃ . . (2.11)The above relations were first obtained by Kuhn [4] and lead to the conclusion that a polymer chaineven if considered as a chain of mutually intersecting steps (i.e. without account of an excludedvolume effect) does not have a shape of a sphere but rather resembles an ellipsoid with unequalaxes . To check how this prediction holds, we have performed numerical simulations of randomwalks on simple cubic lattices, constructing trajectories with the number of steps N up to and performing the averaging over configurations. As one can see from figure 2, the results ofsimulations are in perfect agreement with the data of (2.11). As it was stated in Kuhn’s paper, the most probable shape of a polymer is a bend ellipsoid, of a bean-like shape:“. . . verbogenes Ellipsoids (etwa die Form einer Bohne). . . ” [4]. hapes of macromolecules in good solvents: field theoretical renormalization group approach a) b)
Figure 2.
Ratios r /r (a) and r /r (b) as functions of the chain length, results of computersimulations. Analytic estimates give: r /r ≃ . and r /r ≃ . (see equations (2.5), (2.9),(2.10)). Here it is worth mentioning another far going prediction of the same paper [4]. Discussinghow might the excluded volume effect the polymer size, Kuhn arrives at the relation betweenthe mean square end-to-end distance and the number of monomers which in our notations reads: q h ~R i = ℓN ν with ν = 1 / ǫ . Although this result is suggested on purely phenomenologicalgrounds, its amazing feature is that the power-law form of the dependence is correctly predicted(cf. equation (3.4) from the forthcoming section). Moreover, Kuhn has estimated ǫ by consideringthe excluded volume effect for a -segment chain for which he found an increase of , a resultwe have verified by exact enumerations. Assuming the power law form, ǫ is estimated as: ǫ ≈ . and thus ν ≈ . , which is perfectly confirmed later, e.g. by Flory theory [2], which in d = 3 gives ν = 3 / . The notation for the correction used by Kuhn by coincidence is the same as that usedmuch later in the famous ε -expansion [5] to develop a perturbation theory for calculation of thispower law by means of the renormalization group technique (see equation (4.12) for the first orderresult). Before explaining how this theory is applied to calculate polymer shape characteristics, letus introduce observables in terms of which such description is performed.
3. Description of polymer shape in terms of gyration tensor and combina-tions of its components
Let ~R n = { x n , . . . , x dn } be the position vector of the n th monomer of a polymer chain ( n =1 , . . . , N ). The mean square of the end-to-end distance R e of a chain thus reads: h R i = h| ~R N − ~R | i , (3.1)here and below, h . . . i denotes the averaging over the ensemble of all possible polymer chain con-figurations. The basic shape properties of a specified spatial conformation of the chain can becharacterized [8, 9] in terms of the gyration tensor Q with components: Q ij = 1 N N X n =1 ( x in − x i CM )( x jn − x j CM ) , i, j = 1 , . . . , d, (3.2)with x i CM = P Nn =1 x in /N being the coordinates of the center-of-mass position vector ~R CM .The spread in the eigenvalues λ i of the gyration tensor describes the distribution of monomersinside the polymer coil and thus measures the asymmetry of the molecule; in particular, for asymmetric (spherical) configuration all the eigenvalues λ i are equal, whereas for the so-called . Blavatska, C. von Ferber, Yu. Holovatch a) b) c) Figure 3.
Seen from far away, polymer coil may resemble the objects of different from. Here, wedistinguish sphere-like (a), prolate (b), and oblate (c) conformations. Eigenvalues of correspond-ing gyration tensor (3.2) satisfy: λ ≃ λ ≃ λ (a), λ > λ ≃ λ (b), and λ ≃ λ > λ (c).Correspondingly, asphericity (3.6) and prolateness (3.7) of these conformations satisfy: A d = 0 , S = 0 (a), < A d < , < S (b), < A d < , − / S < (c). See the text for moredetails. prolate and oblate configurations in d = 3 (see figure 3) the eigenvalues satisfy λ ≫ λ ≈ λ and λ ≈ λ ≫ λ correspondingly. Solc and Stockmayer [8] introduced the normalized averageeigenvalues λ i of the gyration tensor as a shape measure of macromolecules. Numerical simulationsin d = 3 dimensions give {h λ i , h λ i , h λ i} = { . , . , . } [16], indicating a high anisotropyof typical polymer configurations compared with the purely isotropic case { / , / , / } .While in simulations the eigenvalues of the gyration tensor Q can easily be calculated andaveraged, different invariants have been devised for theoretical calculations. As far as Q has threeeigenvalues in d = 3 , one may construct three independent combinations of invariants. These arethe square radius of gyration R , the asphericity A d and the prolateness S as elaborated in thefollowing. The first invariant of Q is the squared radius of gyration R = 1 N N X n =1 ( ~R n − ~R CM ) = d X i =1 Q ii = Tr Q , (3.3)which measures the distribution of monomers with respect to the center of mass. To characterizethe size measure of a single flexible polymer chain, one usually considers the mean-squared end-to-end distance h R i (3.1) and radius of gyration h R i (3.3), both governed by the same scalinglaw: h R i ∼ h R i ∼ N ν , (3.4)where N is the mass of the macromolecule (number of monomers in a polymer chain) and ν isa universal exponent ( ν > / d < , ν = 1 / d > ). The ratio of these two characteristicdistances, the so-called size ratio: g d ≡ h R i / h R i , (3.5)appears to be a universal, rotationally-invariant quantity ( g d > d < , g d = 6 ( d > ) [15].Let λ ≡ Tr Q /d be the mean eigenvalue of the gyration tensor. Then one may characterize theextent of asphericity of a polymer chain configuration by the quantity A d defined as [10]: A d = 1 d ( d − d X i =1 ( λ i − λ ) λ = dd − ˆQ (Tr Q ) , (3.6)with ˆQ ≡ Q − λ I (here I is the unity matrix). This universal quantity equals zero for a sphericalconfiguration, where all the eigenvalues are equal, λ i = λ , and takes a maximum value of one in hapes of macromolecules in good solvents: field theoretical renormalization group approach the case of a rod-like configuration, where all the eigenvalues equal zero except one. Thus, theinequality holds: A d . Another rotational invariant quantity, defined in three dimensions,is the so-called prolateness S : S = Q i =1 ( λ i − λ ) λ = 27 det ˆQ (Tr Q ) . (3.7)If the polymer is absolutely prolate, rod-like ( λ = 0 , λ = λ = 0 ), it is easy to see that S equals two. For absolutely oblate, disk-like conformations ( λ = λ , λ = 0 ), this quantity takeson a value of − / , while for a spherical configuration S = 0 . In general, S is positive for prolateellipsoid-like polymer conformations ( λ ≫ λ ≈ λ ) and negative for oblate ones ( λ ≈ λ ≫ λ ),whereas its magnitude measures how much oblate or prolate the polymer is. Note that since λ andthe quantities in (3.3)–(3.7) are expressed in terms of rotational invariants, there is no need toexplicitly determine the eigenvalues λ i which greatly simplifies the calculations.The average of the quantities (3.3)–(3.7) for a given polymer chain length N , denoted as h . . . i ,is performed over an ensemble of possible configurations of a chain. Note that some analyticaland numerical approaches avoid the averaging of the ratio in (3.6), (3.7) and evaluate the ratio ofaverages: ˆ A d = 1 d ( d − d X i =1 h ( λ i − λ ) ih λ i , ˆ S = Q i =1 h ( λ i − λ ) ih λ i , (3.8)which should be distinguished from the averaged asphericity and prolateness: h A d i = 1 d ( d − * d X i =1 ( λ i − λ ) λ + , h S i = * Q i =1 ( λ i − λ ) λ + . (3.9)Contrary to h A d i and h S i , the quantities (3.8) have no direct relation to the probability distributionof the shape parameters A d and S . As pointed out by Cannon [11], this definition overestimates theeffect of larger polymer configurations on the mean shape properties and suppresses the effect ofcompact ones. This artificially leads to overestimated values for shape parameters. The differencebetween h A d i and ˆ A d was found to be great (see table 1). Table 1.
Size ratio, averaged asphericity and prolateness of flexible polymer chains on regulartwo- and three-dimensional lattices. MC: Monte Carlo simulations, DR: direct renormalizationapproaches. a : reference [13], b : [12], c : [17], d : [10], e : [14]. d Method g d h A d i ˆ A d h S i ˆ S . ± . a . ± . b . ± . b – –2 DR . c – . d – . d . ± . e . ± . e . ± . b . ± . e –3 DR . c . e . d – . d Numerous experimental studies indicate that a typical flexible polymer chain in good solventtakes on the shape of an elongated, prolate ellipsoid, similar to what was shown in section 3 for arandom walk. In particular, using the data of x-ray crystallography and cryo-electron microscopy,it was found that the majority of non-globular proteins are characterized by A values from . to . and S values from to . [21, 22]. The shape parameters of polymers were analyzedanalytically, based on the direct renormalization group approach [10, 14, 17], and estimated innumerical simulations [8, 12, 13, 18]. Table 1 gives typical data for the above introduced shapecharacteristics of long flexible polymer chains in d = 2 and d = 3 . . Blavatska, C. von Ferber, Yu. Holovatch Since the above shape and size characteristics of polymer macromolecules are universal, i.e.independent of the details of their chemical structure, they were (along with polymer scalingexponents) the subject of analysis by field-theoretical renormalization group approaches. In thesubsequent section we will introduce this approach as used to calculate polymer shapes.
4. Field-theoretical renormalization group approach to definepolymer shape
Aronovitz and Nelson [10] developed a scheme, allowing one to compute the universal shapeparameters of long flexible polymers within the frames of advanced field theory methods.Here, we start with the Edwards continuous chain model [27], representing the polymer chainby a path ~r ( t ) , parameterized by t T (see figure1 (b)). The system can be described by theeffective Hamiltonian H : H = 12 T Z d t (cid:18) d ~r ( t )d t (cid:19) + u T Z d t T Z d t ′ δ d ( ~r ( t ) − ~r ( t ′ )) . (4.1)The first term in (4.1) represents the chain connectivity, whereas the second term describes theshort range excluded volume interaction with coupling constant u .In this scheme, the gyration tensor components (equation (3.2)) can be rewritten as: Q ij = 12 T T Z d t T Z d t (cid:2) r i ( t ) − r i ( t ) (cid:3) (cid:2) r j ( t ) − r j ( t ) (cid:3) , i, j = 1 , . . . , d. (4.2)The model (4.1) may be mapped to a field theory by a Laplace transform from the Gaussiansurface T to the conjugated chemical potential variable (mass) µ according to [28, 29]: ˆ Z ( µ ) = Z d T exp[ − µ T ] Z ( T ) , (4.3)where Z ( T ) = R D [ r ] exp( −H ) is the partition function of the system as function of the Gaussiansurface and R D [ r ] means an integration over all possible path configurations [30]. Exploiting theanalogy between the polymer problem and O ( m ) symmetric field theory in the limit m → (de Gennes limit) [2], it was shown [28] that the partition function of the polymer system is relatedto the m = 0 -component field theory with an effective Lagrangean: L = Z d d x (cid:20)
12 ( µ | ~φ ( x ) | + |∇ ~φ ( x ) | )+ u
4! ( ~φ ( x )) (cid:21) . (4.4)Here, ~φ is an m -component vector field ~φ = ( φ , . . . , φ m ) and: ˆ Z ( µ ) = Z D [ ϕ ]e −L . (4.5)One of the ways of extracting the scaling behavior of the model (4.4), is to apply the field-theoretical renormalization group (RG) method [5] in the massive scheme, with the Green’s func-tions renormalized at non-zero mass and zero external momenta. The Green’s function G ( N,L )0 canbe defined as an average of N field components and L ϕ -insertions performed with the corre-sponding effective Lagrangean L : δ ( X k i + X p j ) G ( N,L )0 ( { k } ; { p } ; µ ; u ) = Λ Z e i( k i R i + p j r j ) h φ ( r ) . . . φ ( r L ) φ ( R ) . . . φ ( R N ) i L eff d d R . . . d d R N d d r . . . d d r L , (4.6) hapes of macromolecules in good solvents: field theoretical renormalization group approach here { k } = ( k , . . . , k L ) , { p } = ( p , . . . , p N ) are the sets of external momenta and Λ is a cut-off [31]. The renormalized Green’s functions G ( N,L )R are expressed in terms of the bare vertexfunctions as follows: G ( N,L )0 ( { k } ; { p } ; ˆ µ ; u ) = Z N/ φ Z − Lφ G ( N,L )R ( { k } ; { p } ; µ ; u ) , (4.7)where Z φ , Z φ are the renormalizing factors, µ , u are the renormalized mass and couplings.The change of coupling constant u under renormalization defines a flow in parametric space,governed by the corresponding β -function: β u ( u ) = ∂u∂ ln l (cid:12)(cid:12)(cid:12) , (4.8)where l is the rescaling factor and (cid:12)(cid:12) stands for evaluation at fixed bare parameters. The fixedpoints (FP) of the RG transformation are given by the zeroes of the β -function. The stable FP u ∗ ,corresponding to the critical point of the system, is defined as the fixed point where ∂β u ( u ) ∂u | u = u ∗ has a positive real part. The flow of renormalizing factors Z φ , Z φ defines the RG functions γ φ ( u ) , ¯ γ φ ( u ) . These functions, evaluated at the stable accessible FP, allow us to estimate the criticalexponents.Exploiting the perturbation theory expansion in parameter ε = 4 − d (deviation of spacedimension from the upper critical one), one receives within the above described scheme up to thefirst order in ε the well-known results for the fixed points: u ∗ RW = 0 , stable for ε , (4.9) u ∗ SAW = 3 ε , stable for ε > . (4.10)Here, u RW describes the case of simple random walks (idealized polymer chain without any in-trachain interactions), and u SAW is the fixed point, governing the scaling of self-avoiding randomwalks.Evaluating the RG functions γ φ ( u ) and ¯ γ φ ( u ) at the above fixed points, one gets the familiarfirst-order results (see e.g. [5]) for the critical exponents ν and γ , that govern the scaling of thepolymer mean size (3.4) and the number of configurations, correspondingly: ν RW = 12 , γ RW = 1 , (4.11) ν SAW = 12 + ε , γ SAW = 1+ ε . (4.12)Following reference [10], the averaged moments of gyration tensor Q (4.2), which are neededto determine the polymer shape characteristics (3.3), (3.6), (3.7), can be expressed in terms ofrenormalized connected Green’s functions (4.7), in particular: h Q ij i = − (cid:18) T X (cid:19) ν Γ( γ )Γ( γ +2 ν +2) G ij G (2)R (0 , , u ∗ ) , (4.13) h Q ij Q kl i = − (cid:18) T X (cid:19) ν Γ( γ )Γ( γ +4 ν +4) G ij | kl G (2)R (0 , , u ∗ ) . (4.14)Here, the following notations are used: G ij ≡ ∂∂q i ∂∂q j (cid:12)(cid:12)(cid:12) q =0 G (2 , (0 , q, − q ; u ∗ ) , (4.15) G ij | kl = ∂∂q i ∂∂q j ∂∂q k ∂∂q l (cid:12)(cid:12)(cid:12) q =0 G (2 , (0 , q , − q , q , − q ; u ∗ ) , (4.16) ¯ X is non-universal quantity, ν and γ are the critical exponents, ∂/∂q i means differentiation by the i -component of vector q , G (2 , and G (2 , are the renormalized connected Green’s function with . Blavatska, C. von Ferber, Yu. Holovatch (e)(a) (b) (c)(d) Figure 4.
Contributions to the Green’s function G (2 , up to one-loop level. Solid lines denotepropagators µ + k , wavy lines illustrate the insertions of the type ϕ , loops imply integrationover internal momenta. ϕ ( q ) , ϕ ( − q ) and 4 insertions ϕ ( q ) , ϕ ( − q ) , ϕ ( q ) , ϕ ( − q ) respectively, calculated at the fixed point for zero external momenta.The isotropy of the original theory implies, in particular, that h Tr Q i = d h Q ii i , so that: h R i = d h Q ii i . (4.17)The mean-squared end-to-end distance h R i can be expressed as: h R i = − (cid:18) T X (cid:19) ν Γ( γ )Γ( γ +2 ν ) (cid:16) ∇ k G (2)R ( k, − k, u ∗ ) (cid:17) (cid:12)(cid:12)(cid:12) k =0 G (2)R (0 , , u ∗ ) . (4.18)where ∇ k means differentiation over components of external momentum k .One can easily convince oneself, that not-universal quantities cancel when the ratio (3.5) isconsidered: g ≡ h R ih R i = Γ( γ +2 ν +2)Γ( γ +2 ν ) (cid:16) ∇ k G (2)R ( k, − k, u ∗ ) (cid:17) (cid:12)(cid:12)(cid:12) k =0 (cid:16) ∇ q G (2 , (0 , q, − q ; u ∗ ) (cid:17) (cid:12)(cid:12)(cid:12) q =0 . (4.19)In derivation of (4.19) we made use of an obvious relation P di =1 G ii = ∇ G (2 , .Computing the asphericity we follow equation (3.8) considering ˆ A d as the ratio of averages: ˆ A d = dd − h Tr ˆQ ih (Tr Q ) i . (4.20)This definition allows one to directly apply the renormalization group scheme described above, andexpress the ˆ A d in terms of the averaged moments of the gyration tensor (4.13), (4.14): ˆ A d = h Q ii i + d h Q ij i − h Q ii Q jj ih Q ii i + d ( d − h Q ii Q jj i , i = j. (4.21)One can again easily convince oneself that all the non-universal quantities in equations (4.13), (4.14)cancel when calculating (4.21). Note that within the RG approach we will focuss only on twouniversal shape characteristics, namely the size ratio (4.19) and asphericity (4.21), as far as thecalculation of prolateness S is particularly cumbersome.To estimate (4.19), one needs to calculate the Green’s functions G (2 , , presented diagrammati-cally in figure 4. Applying the renormalization procedure, as described above, one finds to the firstorder of the renormalized coupling u (i.e., in the so-called one-loop approximation): G (2 , (0 , q, − q ; u ) = 2 q +1 −
43 1 q +1 uI (0 , q ) − uI (0 , , q )+ 43 1 q +1 uI (0 , , (4.22) hapes of macromolecules in good solvents: field theoretical renormalization group approach (j)(a) (b) (c)(d) (e) (f)(g) (h) (i) Figure 5.
Contributions to the Green’s function G (2 , up to one-loop level. Solid lines denotepropagators µ + k , wavy lines illustrate the insertions of the type ϕ , loops imply integrationover internal momenta. the one-loop integrals I i are listed in appendix A. Performing an ε = 4 − d -expansion of loopintegrals (see appendix A for details) and differentiating over components of vector q , we found: (cid:16) ∇ q G (2 , (0 , q, − q ; u ) (cid:17) (cid:12)(cid:12)(cid:12) q =0 = − − u, (4.23) (cid:16) ∇ k G (2)R ( k, − k ) (cid:17) (cid:12)(cid:12)(cid:12) k =0 = − . (4.24)Substituting the values of fixed points (4.9) and (4.10) into the ratio of these functions, one receivesthe corresponding value of the g -ratio: g RW = 6 , (4.25) g SAW = 6+ ε . (4.26)Equation (4.26) presents a first order correction to the g -ratio caused by excluded volume interac-tions.To compute the averaged asphericity ratio using (4.21), one needs the Green function G (2 , with four insertions ϕ ( q ) / , ϕ ( − q ) / , ϕ ( q ) / , ϕ ( − q ) / , which is schematically presented infigure 5. Applying the renormalization scheme described above, one receives an analytic expressionfor the renormalized function G (2 , , given in the appendix B, equation (B1). Taking derivativesover components of inserted vectors q , q and performing ε -expansions of the resulting expressions(see appendix A for details) we arrive at the following expansions for the functions (4.15), (4.16): G xx = 576+ 4028 u ,G xx | yy = 320+ 436 u ,G xy | xy = 128+ 308 u . (4.27)Evaluating these relations at the fixed points (4.9), (4.10) and substituting into (4.13), (4.14), . Blavatska, C. von Ferber, Yu. Holovatch (4.21) one receives: ˆ A RW d = 12 , (4.28) ˆ A SAW d = 12 + 15512 ε . (4.29)We note, that result (4.28) means that even within the idealized model of simple random walks, theshape of a polymer chain is highly anisotropic (cf. Kuhn’s picture, described in section 2). Takinginto account the excluded volume effect makes the polymer chain more extended and aspherical:indeed, in three dimensions ( ε = 1 ) the above obtained quantity reads: ˆ A SAW d ≃ . .
5. Polymer in porous environment: Model with long-range correlated disor-der
In real physical processes, one is often interested how structural obstacles (impurities) in theenvironment alter the behavior of a system. The density fluctuations of obstacles lead to a largespatial inhomogeneity and create pore spaces, which are often of fractal structure [32]. In polymerphysics, it is of great importance to understand the behavior of macromolecules in the presenceof structural disorder, e.g., in colloidal solutions [33] or near the microporous membranes [34]. Inparticular, a related problem concerns the protein folding dynamics in the cellular environment,which can be considered as a highly disordered environment due to the presence of a large amountof biochemical species, occupying up to of the total volume [35]. Structural obstacles stronglyaffect the protein folding [36]. Recently, it was realized experimentally [37] that macromolecularcrowding has a dramatic effect on the shape properties of proteins.In the language of lattice models, a disordered environment with structural obstacles can beconsidered as a lattice, where some amount of randomly chosen sites contain defects which areto be avoided by the polymer chain. Of particular interest is the case when the concentrationof lattice sites allowed for the SAWs equals the critical concentration and the lattice is at thepercolation threshold. In this regime, SAWs belong to a new universality class, the scaling law (3.4)holds with exponent ν p c > ν SAW [38]. The universal shape characteristics of flexible polymers indisordered environments modeled by a percolating lattice were studied recently in [39]. Anotherinteresting situation arises when the structural obstacles of environment display correlations ona mesoscopic scale [40]. One can describe such a medium by a model of long-range-correlated(extended) quenched defects. This model was proposed in reference [41] in the context of magneticphase transitions. It considers defects, characterized by a pair correlation function h ( r ) , that decayswith a distance r according to a power law: h ( r ) ∼ r − a (5.1)at large r . This type of disorder has a direct interpretation for integer values of a ; namely, thecase a = d corresponds to point-like defects, while a = d − a = d − describes straight lines(planes) of impurities of random orientation. Non-integer values of a are interpreted in terms ofimpurities organized in fractal structures [42]. The effect of this type of disorder on the magneticphase transitions has been a subject to numerous studies [43].The impact of long-range-correlated disorder on the scaling of single polymer chains was ana-lyzed in our previous works [44] by means of field-theoretical renormalization group approach. Inparticular, it was shown that the correlated obstacles in environment lead to a new universalityclass with values of the polymer scaling exponents that depend on the strength of correlation ex-pressed by parameters a . The question about how the characteristics of shape of a flexible chainare effected by the presence of such a porous medium was briefly discussed by us in reference [45].The details of these calculations will be presented here.We introduce disorder into the model (4.4), by redefining µ → µ + δµ ( x ) , where the local hapes of macromolecules in good solvents: field theoretical renormalization group approach fluctuations δµ ( x ) obey: hh δµ ( x ) ii = 0 , hh δµ ( x ) δ ˆ µ ( y ) ii = h ( | x − y | ) . (5.2)Here, hh . . . ii denotes the average over spatially homogeneous and isotropic quenched disorder. Theform of the pair correlation function h ( r ) is chosen to decay with distance according to (5.1).In order to average the free energy over different configurations of the quenched disorder weapply the replica method to construct an effective Lagrangean [44]: L dis = 12 n X α =1 Z d d x (cid:20)(cid:16) µ | ~φ α ( x ) | + |∇ ~φ α ( x ) | (cid:17) + u (cid:16) ~φ α ( x ) (cid:17) (cid:21) + n X α,β =1 Z d d x d d yh ( | x − y | ) ~φ α ( x ) ~φ β ( y ) . (5.3)Here, the term describing replicas coupling contains the correlation function h ( r ) (5.1), Greekindices denote replicas and both the replica ( n → ) and polymer ( m → ) limits are implied. Forsmall k , the Fourier-transform ˜ h ( k ) of (5.1) reads: ˜ h ( k ) ∼ v + w | k | a − d . (5.4)Taking this into account, rewriting equation (5.3) in momentum space variables, and recallingthe special symmetry properties of (5.3) that appear for m , n → [44], a theory with two barecouplings u , w results. Note that for a > d the w -term is irrelevant in the RG sense and onerestores the pure case (absence of structural disorder). As it will be shown below, this term modifiesthe critical behaviour at a < d . We will refer to this type of disorder as long-range-correlated anddenote by LR hereafter.To extract the scaling behavior of the model (5.3), one applies the field-theoretical renormaliza-tion group method following the scheme described in a previous section, with modifications causedby the presence of the second coupling constant w . In particular, the change of couplings u , w under renormalization defines a flow in parametric space, governed by corresponding β -functions(c.f. equation (4.8)): β u ( u, w ) = ∂u∂ ln l (cid:12)(cid:12)(cid:12) , β w ( u, w ) = ∂w∂ ln l (cid:12)(cid:12)(cid:12) . (5.5)The fixed points (FPs) of the RG transformation are given by common zeroes of the β -functions.The stable FP ( u ∗ , w ∗ ) that corresponds to the critical point of the system, is defined as the fixedpoint where the stability matrix B ij = ∂β λ i /∂λ j , i, j = 1 , possesses eigenvalues with positivereal parts (here, λ = u , λ = w ).In our previous work [44] we have found the FP coordinates for polymers in LR disorder, whichup to the first order of ε = 4 − d , δ = 4 − a -expansion read: u ∗ RW = 0 , w ∗ RW = 0 stable for δ < , δ < , (5.6) u ∗ SAW = 3 ε , w ∗ SAW = 0 stable for δ < ε/ , (5.7) u ∗ LR = 3 δ ε − δ ) , w ∗ LR = 3 δ ( ε − δ )2( ε − δ ) stable for ε/ < δ < ε. (5.8)The RW and SAW fixed points restore the corresponding cases of a polymer in pure solvent(cf. (4.9), (4.10)), whereas the LR fixed point reflects the effect of correlated obstacles, whichappears to be non-trivial in certain regions of the d , a plane and to govern new scaling behaviourin this region. For critical exponents ν LR , γ LR governing the scaling of polymer chains in the regionof a , d , where the effect of LR disorder is nontrivial, one obtains [44]: ν LR = 1 / δ/ , γ LR = 1+ δ/ . (5.9) . Blavatska, C. von Ferber, Yu. Holovatch As it was explained in the previous section, to estimate the universal size ratio (4.19) for thecase of polymers in long-range-correlated disorder, we calculate the Green function G (2 , ( u, w ) ,presented diagrammatically in figure 4. Now, every interactive diagram appears twice, once witheach of the two couplings u and w , respectively. For the renormalized function we obtain up to theone-loop approximation: G (2 , (0 , q, − q ; u, w ) = 2 q +1 −
43 1 q +1 [ uI (0 , q ) − wJ (0 , q )] −
23 [ uI (0 , , q ) − wJ (0 , , q )] + 43 1 q +1 [ uI (0 , − wJ (0 , . (5.10)The one-loop integrals I i are given in appendix A. Note that J i differs from corresponding I i only by an additional factor | p | a − d in the numerator. Differentiating over components of vector q ,evaluating at q = 0 and performing double ε, δ -expansions of the resulting expression, we have: (cid:16) ∇ q G (2 , (0 , q, − q ; u, w ) (cid:17) (cid:12)(cid:12)(cid:12) q =0 = − − u − w, (5.11) (cid:16) ∇ k G (2)R ( k, − k ) (cid:17) (cid:12)(cid:12)(cid:12) k =0 = − . (5.12)Recalling the value of the LR fixed point (5.8), and inserting this into the ratio (4.19) of thesefunctions, one arrives at an expansion for the size ratio of a SAW in the presence of long-rangecorrelated disorder: g LR = 6+ δ . (5.13)This should be compared with the corresponding value in the pure case (4.26). Let us qualitativelyestimate the change in the size ratio g , caused by the presence of structural obstacles in threedimensions. Substituting directly ε = 1 into (4.26), we have for the polymer chain in a puresolvent: g pure ≃ . . Let us recall that the effect of long-range-correlated disorder is relevant to a d ( δ > ε ) (see e.g. explanation after equation (5.3)). Estimates of g LR can be evaluated bydirect substitution of the continuously variable parameter δ into equation (5.13). One concludesthat increasing the parameter δ (which corresponds to an increase of disorder strength) leads to acorresponding increase of the g -ratio.To compute the averaged asphericity in correspondence with (4.21), we need the Green function G (2 , ( u, w ) with four insertions ϕ ( q ) / , ϕ ( − q ) / , ϕ ( q ) / , ϕ ( − q ) / , shown diagrammati-cally in figure 5. The corresponding analytic expression for the renormalized function is given inappendix B, equation (B2). Taking derivatives of this expression with respect to the componentsof the inserted vectors q , q we find: G xx = 576+ 402815 ( u − w ) ,G xx | yy = 320+ 4363 ( u − w ) ,G xy | xy = 128+ 3085 ( u − w ) . (5.14)At the LR fixed point (5.8), from (4.21) we finally have: ˆ A LR d = 12 + 148 ε + 13768 δ. (5.15)Again, let us qualitatively estimate the change in ˆ A d caused by the presence of structural obstaclesin three dimensions. Substituting directly ε = 1 into (4.29), we have for the pure case: ˆ A pure d ≃ . .Estimates of ˆ A LR d can be obtained by direct substitution of the continuously changing parameter δ into equation (5.15). An increasing strength of disorder correlations results in an increase of theasphericity ratio of polymers in disorder. This phenomenon is intuitively understandable if one re-calls the impact of the long-range-correlated disorder on the mean end-to-end distance exponent ν . hapes of macromolecules in good solvents: field theoretical renormalization group approach Indeed, it has been shown in [44] that such disorder leads to an increase of ν , and subsequently,to the swelling of a polymer chain. Extended obstacles do not favour return trajectories and as aresult the polymer chain becomes more elongated. In turn, such elongation leads to an increase ofthe asphericity ratio as predicted by equation (5.15).
6. Conclusions
The universal characteristics of the average shape of polymer coil configurations in a porous(crowded) environment with structural obstacles have been analyzed considering the special casewhen the defects are correlated at large distances r according to a power law: h ( r ) ∼ r − a . Applyingthe field-theoretical RG approach, we estimate the size ratio g = h R i / h R i and averaged aspheric-ity ratio ˆ A d up to the first order of a double ε = 4 − d , δ = 4 − a expansion. We have revealed thatthe presence of long-range-correlated disorder leads to an increase of both g and ˆ A d as comparedto their values for a polymer chain in a pure solution. Moreover, the value of the asphericity ratio ˆ A d was found to be closer to the maximal value of one in presence of correlated obstacles. Thus,we conclude that the presence of structural obstacles in an environment enforces the polymer coilconfigurations to be less spherical. We believe that the obtained first order results indicate theappropriate qualitative changes in polymer shape caused by long-range-correlated environment.However, to get more accurate quantitative results, higher orders of perturbation theory may beneeded. This is subject of further investigations. Acknowledgements
We thank Prof. Myroslav Holovko (Lviv) for an invitation to submit a paper to this Festschriftand Prof. Yuri Kozitsky (Lublin) for useful discussions. This work was supported in part by theFP7 EU IRSES project N269139 “Dynamics and Cooperative Phenomena in Complex Physicaland Biological Media” and Applied Research Fellowship of Coventry University.
Appendix A
Here, we present the expressions for the loop integrals, as they appear in the Green functions G (2 , and G (2 , . We make the couplings dimensionless by redefining u = u µ d − and w = w µ a − ;therefore, the loop integrals do not explicitly contain the mass [31]: I ( k , k ) = Z d ~p [( p + k ) +1] [( p + k ) +1] ,I ( k , k , k ) = Z d ~p [( p + k ) +1][( p + k ) +1][( p + k ) +1] ,I ( k , k , k , k ) = Z d ~p [( p + k ) +1][( p + k ) +1][( p + k ) +1][( p + k ) +1] ,I ( k , k , k , k , k ) = Z d ~p [( p + k ) +1][( p + k ) +1][( p + k ) +1][( p + k ) +1][( p + k ) +1] . The loop integrals J i (equations (5.10) and (B2)) differ from corresponding I i only by an additionalfactor | p | a − d in the numerators. The correspondence of the integrals to the diagrams in figure 5 is: ( e ) , ( f ) : integrals I (0 , q ) , I (0 , q ) , J (0 , q ) , J (0 , q ) , ( g ) : I (0 , q , q ) , I (0 , , q ) , I (0 , , q ) , J (0 , q , q ) , J (0 , , q ) , J (0 , , q ) ,J (0 , q , q ) , J (0 , , q ) , J (0 , , q ) , ( h ) : I (0 , , q ) , I (0 , , q ) , I (0 , q , q + q ) , I (0 , q , q + q ) , J (0 , , q ) ,J (0 , , q ) , J (0 , q , q + q ) , J (0 , q , q + q ) , . Blavatska, C. von Ferber, Yu. Holovatch ( i ) : I (0 , , q , q ) , I (0 , q , q , q + q ) , I (0 , q , q , q + q ) , I (0 , q , q , q + q ) ,J (0 , , q , q ) , J (0 , q , q , q + q ) , J (0 , q , q , q + q ) , J (0 , q , q , q + q ) , ( j ) : I (0 , , , q , q ) , I (0 , , q , q , q + q ) , I (0 , , q , q , q + q ) , I (0 , , q , q , q + q ) ,J (0 , , , q , q ) , J (0 , , q , q , q + q ) , J (0 , , q , q , q + q ) , J (0 , , q , q , q + q ) . In our calculations, we use the following formula to fold many denominators into one (see e.g.book of D. Amit in reference [5]): a α . . . a α n n = Γ( α + . . . + α n )Γ( α ) . . . Γ( α n ) ×× Z d x . . . Z d x n − x α n − . . . x α n − − n − (1 − x − . . . − x n − ) α n − [ x a + . . . + x n − a n − +(1 − x − . . . − x n − ) a n ] α + ... + α n , (A1)where the Feynmann variables x i extend over the domain x + . . . + x n − .To compute the d -dimensional integrals we apply : Z d p ( p +2 ~k~p + m ) α = 12 Γ( d/ α − d/ α ) ( m − k ) d/ − α , (A2)here d p = d d p Ω d / (2 π ) d , where the geometrical angular factor Ω d = 1 / (2 d − π d/ Γ( d/ is sepa-rated out and absorbed by redefining the coupling constant.As an example we present the calculation of the integral: I (0 , q , q ) = Z d ~p ( p +1)[( p + q ) +1][( p + q ) +1] . (A3)First, we make use of formula (A1) to rewrite: p +1)[( p + q ) +1][( p + q ) +1] = Γ(3) R d x R d x [ p +2 ~p ( ~q x + ~q x )+1+ x q + x q ] . Now one can perform the integration over p , passing to the d -dimensional polar coordinates andmaking use of the formula (A2): Z d p [ p +2 ~p ( ~q x + ~q x )+1+ x q + x q ] =Γ( d/ − d/ x q (1 − x )+ x q (1 − x ) − x x q q ] d/ − . (A4)As a result, we are left with: I (0 , q , q ) = 12 Γ (cid:18) d (cid:19) Γ (cid:18) − d (cid:19) ×× Z d x Z d x [1+ x q (1 − x )+ x q (1 − x ) − x x q q ] d/ − . (A5)To find the contributions of this integral to G xx | xx and G xx | yy (according to (4.16)), we firstdifferentiate the integrand in (A5) over the components of vectors ~q , ~q : I xx | xx ≡ dd q x dd q x dd q x dd q x (cid:2) x q (1 − x )+ x q (1 − x ) − x x q q (cid:3) d/ − (cid:12)(cid:12)(cid:12) ~q = ~q =0 = 4 (cid:18) d − (cid:19) (cid:2) x x + x x (1 − x )(1 − x ) (cid:3) − (cid:18) d − (cid:19) (cid:2) x x (1 − x )(1 − x )+4 x x (cid:3) ,I xx | yy ≡ dd q x dd q x dd q y dd q y (cid:2) x q (1 − x )+ x q (1 − x ) − x x q q (cid:3) d/ − | ~q = ~q =0 = (cid:18) d − (cid:19) x x (1 − x )(1 − x ) − (cid:18) d − (cid:19) x x (1 − x )(1 − x ) . hapes of macromolecules in good solvents: field theoretical renormalization group approach Finally, the contributions of I (0 , q , q ) to G xx | xx and G xx | yy are found by performing integrationof I xx | xx and I xx | yy over x , x in (A5). Results can further be evaluated either fixing the valueof space dimension d or performing an expansion in parameter ε = 4 − d . Working within the lastapproach, we obtain up to the first order of the ε -expansion:
12 Γ (cid:18) d (cid:19) Γ (cid:18) − d (cid:19) Z d x Z d x I xx | xx ≃
19 + 136 ε,
12 Γ (cid:18) d (cid:19) Γ (cid:18) − d (cid:19) Z d x Z d x I xx | yy ≃
1+ 14 ε. (A6) Appendix B
In this appendix we give the expressions for renormalized Green functions G (2 , with fourinsertions ϕ ( q ) / , ϕ ( − q ) / , ϕ ( q ) / , ϕ ( − q ) / one needs for calculation of the averagedasphericity ratios ˆ A SAW d , ˆ A LR d defined by (4.21). The Green function G (2 , ( u ) (see figure 5) reads: G (2 , ( u ) = 8( q +1)( q +1) + 8( q +1)( q +1)[( q + q ) +1] + 4( q +1) + 4( q +1) − (cid:26) u [2 I (0 , q )+2 I (0 , q )]( q +1)( q +1)[( q + q ) +1] − u [2 I (0 , q )+2 I (0 , q )+2 I (0 , q , q )]( q +1)( q +1) − uI (0 , q )( q +1) [( q + q ) +1] − uI (0 , q )( q +1) [( q + q ) +1] − uI (0 , , q )( q +1) − uI (0 , , q )( q +1) − u [ I (0 , , q )+ I (0 , , q , q )+ I (0 , q , q , q + q )+ I (0 , q , q , q + q )]( q +1) − u [ I (0 , , q )+ I (0 , , q , q )+ I (0 , q , q , q + q )+ I (0 , q , q , q + q )]( q +1) − u [ I (0 , q , q + q )+ I (0 , q , q + q )]( q +1)[( q + q ) +1] − u [ I (0 , q , q + q )+ I (0 , q , q + q )]( q +1)[( q + q ) +1] − u [2 I (0 , , , q , q )+2 I (0 , , q , q , q + q )+ I (0 , , q , q , q + q )+ I (0 , , q , q , q + q )]2 (cid:27) + 4 uI (0 , (cid:20) q +1)( q +1) + 8( q +1)( q +1)[( q + q ) +1] + 4( q +1) + 4( q +1) (cid:21) . (B1)The Green function G (2 , ( u, w ) reads: G (2 , ( u, w ) = 8( q +1)( q +1) + 8( q +1)( q +1)[( q + q ) +1] + 4( q +1) − (cid:26) u [2 I (0 , q )+2 I (0 , q )]( q +1)( q +1)[( q + q ) +1] − u [2 I (0 , q )+2 I (0 , q )+2 I (0 , q , q )]( q +1)( q +1) − w [2 J (0 , q )+2 J (0 , q )]( q +1)( q +1)[( q + q ) +1] − w [2 J (0 , q )+2 J (0 , q )+2 J (0 , q , q )]( q +1)( q +1) − uI (0 , q )( q +1) [( q + q ) +1] − uI (0 , q )( q +1) [( q + q ) +1] − uI (0 , , q )( q +1) − uI (0 , , q )( q +1) − wJ (0 , q )( q +1) [( q + q ) +1] − wJ (0 , q )( q +1) [( q + q ) +1] − wJ (0 , , q )( q +1) − wJ (0 , , q )( q +1) − u [ I (0 , , q )+ I (0 , , q , q )+ I (0 , q , q , q + q )+ I (0 , q , q , q + q )]( q +1) . Blavatska, C. von Ferber, Yu. Holovatch − w [ J (0 , , q )+ J (0 , , q , q )+ J (0 , q , q , q + q )+ J (0 , q , q , q + q )]( q +1) − u [ I (0 , , q )+ I (0 , , q , q )+ I (0 , q , q , q + q )+ I (0 , q , q , q + q )]( q +1) − w [ J (0 , , q )+ J (0 , , q , q )+ J (0 , q , q , q + q )+ J (0 , q , q , q + q )]( q +1) − u [ I (0 , q , q + q )+ I (0 , q , q + q )]( q +1)[( q + q ) +1] − u [ I (0 , q , q + q )+ I (0 , q , q + q )]( q +1)[( q + q ) +1] − w [ J (0 , q , q + q )+ J (0 , q , q + q )]( q +1)[( q + q ) +1] − w [ J (0 , q , q + q )+ J (0 , q , q + q )]( q +1)[( q + q ) +1] − u [2 I (0 , , , q , q )+2 I (0 , , q , q , q + q )+ I (0 , , q , q , q + q )+ I (0 , , q , q , q + q )]2 − w [2 J (0 , , , q , q )+2 J (0 , , q , q , q + q )+ J (0 , , q , q , q + q )+ J (0 , , q , q , q + q )]2 (cid:27) + 4[ uI (0 , − wJ (0 , (cid:20) q +1)( q +1) + 8( q +1)( q +1)[( q + q ) +1] + 4( q +1) + 4( q +1) (cid:21) . (B2)The one-loop integrals I i and J i are explained in appendix A. References
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A, 2010, Форми макромолекул у хороших розчинниках:пiдхiд теоретико-польової ренормалiзацiйної групи
В. Блавацька , К. фон Фербер , Ю. Головач Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна Дослiдницький центр прикладної математики, Унiверситет Ковентрi, CV1 5FB Ковентрi, Англiя Теоретична фiзика полiмерiв, Унiверситет Фрайбургу, D-79104 Фрайбург, НiмеччинаУ статтi ми показуємо, яким чином можна застосувати метод теоретико-польової ренормалiзацiйноїгрупи для аналiзу унiверсальних властивостей форм довгих гнучких полiмерних ланцюгiв у пористомусередовищi. До цього часу такi аналiтичнi розрахунки в основному торкались показникiв скейлiнгу, щовизначають конформацiйнi властивостi полiмерних макромолекул. Проте, iснують й iншi спостережува-нi величини, що, як i показники скейлiнгу, є унiверсальними (тобто незалежними вiд хiмiчної структурияк макромолекул, так i розчинника), а отже можуть бути проаналiзованi в межах пiдходу ренормалi-зацiйної групи. Ми цiкавимось питанням, якої форми набуває довга гнучка полiмерна макромолекулау розчинi в присутностi пористого середовища. Це питання є суттєвим для розумiння поведiнки ма-кромолекул у колоїдних розчинах, поблизу мiкропористих мембран, а також у клiтинному середовищi.Ми розглядаємо запропоновану ранiше модель полiмера у d -вимiрному просторi [V. Blavats’ka, C. vonFerber, Yu. Holovatch, Phys. Rev. E, 2001, , 041102] у середовищi iз структурними неоднорiдностями,що характеризуються парною кореляцiйною функцiєю h ( r ) , яка спадає iз вiдстанню r згiдно степене-вого закону: h ( r ) ∼ r − a . Застосовуємо пiдхiд теоретико-польової ренормалiзацiйної групи i оцiнюємовiдношення розмiрiв h R i / h R i та асферичнiсть ˆ A d до першого порядку ε = 4 − d , δ = 4 − a -розкладу. Ключовi слова: полiмер, заморожений безлад, ренормалiзацiйна групаполiмер, заморожений безлад, ренормалiзацiйна група